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Mirrors > Home > ILE Home > Th. List > tgidm | Unicode version |
Description: The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgidm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgvalex 11917 |
. . . . 5
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2 | eltg3 11924 |
. . . . 5
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3 | 1, 2 | syl 14 |
. . . 4
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4 | uniiun 3805 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | simpr 109 |
. . . . . . . . . . . . 13
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6 | 5 | sselda 3039 |
. . . . . . . . . . . 12
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7 | eltg4i 11922 |
. . . . . . . . . . . 12
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8 | 6, 7 | syl 14 |
. . . . . . . . . . 11
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9 | 8 | iuneq2dv 3773 |
. . . . . . . . . 10
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10 | 4, 9 | syl5eq 2139 |
. . . . . . . . 9
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11 | iuncom4 3759 |
. . . . . . . . 9
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12 | 10, 11 | syl6eq 2143 |
. . . . . . . 8
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13 | inss1 3235 |
. . . . . . . . . . . 12
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14 | 13 | rgenw 2441 |
. . . . . . . . . . 11
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15 | iunss 3793 |
. . . . . . . . . . 11
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16 | 14, 15 | mpbir 145 |
. . . . . . . . . 10
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17 | 16 | a1i 9 |
. . . . . . . . 9
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18 | eltg3i 11923 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | sylan2 281 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 12, 19 | eqeltrd 2171 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | eleq1 2157 |
. . . . . . 7
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22 | 20, 21 | syl5ibrcom 156 |
. . . . . 6
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23 | 22 | expimpd 356 |
. . . . 5
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24 | 23 | exlimdv 1754 |
. . . 4
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25 | 3, 24 | sylbid 149 |
. . 3
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26 | 25 | ssrdv 3045 |
. 2
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27 | bastg 11928 |
. . 3
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28 | tgss 11930 |
. . 3
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29 | 1, 27, 28 | syl2anc 404 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 26, 29 | eqssd 3056 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-topgen 11840 |
This theorem is referenced by: tgss3 11945 |
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